The motivation for this paper is the construction of entropy stable schemes for
the Euler equations with gravity. These equations are of fundamental importance
in simulations of astrophysical phenomena,
atmospheric large-eddy simulations,
numerical weather prediction, and climate modeling.
Due to their excellent properties, DG schemes are already emerging
in computational astrophysics [1]}, [2]} as well as
in small- and large-scale atmospheric modeling
[3]}, [4]}, [5]}, [6]}.
In contrast to engineering applications,
many atmospheric models formulate the equations in term of
an entropy-type variable known as potential temperature,
which has favorable properties for simulating atmospheric flows [7]}.
However, in climate modeling applications,
conservation of total energy and balanced energy transfers are also of great interest.
This can be achieved, for example, by using mimetic methods [8]}.
An alternative, that we propose in this paper, is to guarantee conservation
of energy by using the total energy formulation, and
use the entropy-stable approach for robustness and consistency
with the second law of thermodynamics.
Crucially, this requires an extension of the existing entropy-stable framework,
since in the total energy formulation of the Euler equations
the gravity force enters as a non-conservative and spatially varying source
term.
